#7816048

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(9,10,13,12)(11,15,16,14)(17,18)(19,27)(20,21)(22,25)(23,26)(24,28)(29,32)(30,31);(9,11,13,16)(10,14,12,15)(17,19,23,28)(18,24,26,27)(20,29,30,25)(21,22,31,32);(1,2,4,6)(3,5,8,7)(19,28)(22,32)(24,27)(25,29);(1,2)(3,5)(4,6)(7,8)(9,12,13,10)(11,15,16,14)(17,20,23,30)(18,21,26,31)(19,29,28,25)(22,27,32,24);(1,3)(2,5)(4,8)(6,7)(17,21)(18,20)(19,22)(23,31)(24,29)(25,27)(26,30)(28,32);(17,22)(18,25)(19,21)(20,27)(23,32)(24,30)(26,29)(28,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7723244);